Essential orders on stratified algebras with duality and $\mathcal{S}$-subcategories in $\mathcal{O}$

Abstract: We prove uniqueness of the essential order for stratified algebras having simple preserving duality, generalizing a recent result of Coulembier for quasi-hereditary algebras. We apply this to classify, up to equivalence, regular integral blocks of $\mathcal{S}$-subcategories in the BGG category $\mathcal{O}$. We also describe various homological invariants of these blocks.